Principles of Modern Radar - Volume 2 1891121537

604 CHAPTER 13 Introduction to Radar Polarimetrythe circular basis set (13.30). If a linear basis is used, the scattering matrix is given by[ ]SHH SS L =VH(13.31)S HV S VVand a similar representation follows in terms of the circular basis. For a reciprocal medium(e.g., no Faraday rotation of polarization), S VH = S HV . It is possible to transform thescattering matrix from the linear basis to the circular basis usingS C = T † LC S LT LC (13.32)where the matrix T LC is defined in (13.29), and the superscript † denotes conjugatetranspose of the matrix. In (13.31), S rt , r, t = V or H, corresponds to the complexscattering amplitude with polarisation t on transmit and polarisation r on receive; forco-polarized channels r = t, and for cross-polarized channels r ≠ t. The absolute phaseof the matrix S does not affect either the wave polarization state or the received signalpower. Therefore, one of the elements in S (say, S HH ) may be used as phase reference forthe other elements, thus resulting in a target scattering matrix with relative phaseS SMR = Se jφ HH(13.33)Without specific reference to (13.33), the relative scattering matrix will be assumed in theremainder of this section.13.3.2 Optimal PolarizationsThe transmit polarization which maximizes or minimizes the total backscattered powercan be determined by solving the pseudo-eigenvalue problem [14]Sê i = λ i ê ∗ i , i = 1, 2 (13.34)where ê 1 and ê 2 are two distinct orthogonal eigenvectors (unit magnitude) associatedwith complex eigenvalues λ 1 and λ 2 , respectively. It can be shown that the eigenvectorscorresponding to orthogonal elliptically polarized states are given by (see (13.20)) [11][ ][ ] [[ ] cos ψ − sin ψ cos τ j sin τ [ ] = mê1 ê 2 = êh êsin ψ cos ψ j sin τ cos τv−n ∗ ] n [m ∗ êh]ê v(13.35)The matrix in (13.35) transforms the linear polarization basis into an elliptical polarizationbasis. It is evident from the second equality that ê 1·ê ∗ 2 = 0, ∣ ê 1 · ê ∗ 1 = 1, and ∣ ê 2 · ê ∗ 2 = 1.Also, ê 2 (ψ,τ) = ê 1 (ψ + π/2, −τ). It then follows that the matrix U = [ ] ê 1 ê 2is unitary, that is, U T U ∗ = I, where I is the identity matrix, and superscript T denotesthe transpose. The unitary transformation U T SU diagonalizes the scattering matrix,resulting in[U T λ1SU =0]0 = Sλ d2(13.36)Inverting (13.36) using the property U −1 = U † , where U † denotes the Hermitian adjointor conjugate transpose, one obtains the scattering matrix from the eigendecomposition